Homework 4 Solutions

Transcription

1 Homework 4 Solutions Problem 1 # Read in Data school.1 <- read.table("~/dropbox/uw/teaching/cs&ss-stat564/solutions/hw4/school1.dat")[, 1] school.2 <- read.table("~/dropbox/uw/teaching/cs&ss-stat564/solutions/hw4/school2.dat")[, 1] school.3 <- read.table("~/dropbox/uw/teaching/cs&ss-stat564/solutions/hw4/school3.dat")[, 1] # Create Functions for Obtaining Posterior Parameters get.kappa.n <- function(y, kappa.0) { kappa.n <- kappa.0 + length(y) return(kappa.n) get.mu.n <- function(y, mu.0, kappa.0, kappa.n) { mu.n <- (kappa.0*mu.0 + sum(y))/kappa.n return(mu.n) get.nu.n <- function(y, nu.0) { nu.n <- nu.0 + length(y) return(nu.n) get.sigma.sq.n <- function(y, mu.0, nu.0, sigma.sq.0, nu.n, kappa.n) { sigma.sq.n <- (nu.0*sigma.sq.0 + sum((y - mean(y))^2) + kappa.0*length(y)*(mean(y) - mu.0)^2/kappa.n)/nu.n return(sigma.sq.n) # Set hyperparameters mu.0 <- 5 sig.sq.0 <- 4 kappa.0 <- 1 nu.0 <- 2 # Compute posterior parameters kappa.n.1 <- get.kappa.n(school.1, kappa.0) kappa.n.2 <- get.kappa.n(school.2, kappa.0) kappa.n.3 <- get.kappa.n(school.3, kappa.0) mu.n.1 <- get.mu.n(school.1, mu.0, kappa.0, kappa.n.1) mu.n.2 <- get.mu.n(school.2, mu.0, kappa.0, kappa.n.2) mu.n.3 <- get.mu.n(school.3, mu.0, kappa.0, kappa.n.3) 1

2 nu.n.1 <- get.nu.n(school.1, nu.0) nu.n.2 <- get.nu.n(school.2, nu.0) nu.n.3 <- get.nu.n(school.3, nu.0) sig.sq.n.1 <- get.sigma.sq.n(school.1, mu.0, nu.0, sig.sq.0, nu.n.1, kappa.n.1) sig.sq.n.2 <- get.sigma.sq.n(school.2, mu.0, nu.0, sig.sq.0, nu.n.2, kappa.n.2) sig.sq.n.3 <- get.sigma.sq.n(school.3, mu.0, nu.0, sig.sq.0, nu.n.3, kappa.n.3) # Take MC Samples S < sigma.sq.inv.samps.1 <- rgamma(s, nu.n.1/2, nu.n.1*sig.sq.n.1/2) theta.samps.1 <- rnorm(s, mu.n.1, sqrt((1/sigma.sq.inv.samps.1)/kappa.n.1)) tilde.y.samps.1 <- rnorm(s, theta.samps.1, sqrt((1/sigma.sq.inv.samps.1))) sigma.sq.inv.samps.2 <- rgamma(s, nu.n.2/2, nu.n.2*sig.sq.n.2/2) theta.samps.2 <- rnorm(s, mu.n.2, sqrt((1/sigma.sq.inv.samps.2)/kappa.n.2)) tilde.y.samps.2 <- rnorm(s, theta.samps.2, sqrt((1/sigma.sq.inv.samps.2))) sigma.sq.inv.samps.3 <- rgamma(s, nu.n.3/2, nu.n.3*sig.sq.n.3/2) theta.samps.3 <- rnorm(s, mu.n.3, sqrt((1/sigma.sq.inv.samps.3)/kappa.n.3)) tilde.y.samps.3 <- rnorm(s, theta.samps.3, sqrt((1/sigma.sq.inv.samps.3))) Part a. summ <- rbind(c(mean(theta.samps.1), quantile(theta.samps.1, c(0.025, 0.975)), mean(sqrt(1/sigma.sq.inv.samps.1)), quantile(sqrt(1/sigma.sq.inv.samps.1), c(0.025, 0.975))), c(mean(theta.samps.2), quantile(theta.samps.2, c(0.025, 0.975)), mean(sqrt(1/sigma.sq.inv.samps.2)), quantile(sqrt(1/sigma.sq.inv.samps.2), c(0.025, 0.975))), c(mean(theta.samps.3), quantile(theta.samps.3, c(0.025, 0.975)), mean(sqrt(1/sigma.sq.inv.samps.3)), quantile(sqrt(1/sigma.sq.inv.samps.3), c(0.025, 0.975)))) row.names(summ) <- 1:3 colnames(summ) <- c("$\\mathbb{e\\left[\\theta \\boldsymbol y\\right]$", "$\\theta \\boldsymbol y_{\\left(0.025\\right)$", "$\\theta \\boldsymbol y_{\\left(0.975\\right)$", "$\\mathbb{e\\left[\\sigma \\boldsymbol y\\right]$", "$\\sigma \\boldsymbol y_{\\left(0.025\\right)$", "$\\sigma \\boldsymbol y_{\\left(0.975\\right)$") kable(summ, digits = 2) 2

3 E [θ y] θ y (0.025) θ y (0.975) E [σ y] σ y (0.025) σ y (0.975) Part b. comp.theta <- rbind(mean(theta.samps.1 < theta.samps.2 & theta.samps.2 < theta.samps.3), mean(theta.samps.1 < theta.samps.3 & theta.samps.3 < theta.samps.2), mean(theta.samps.2 < theta.samps.1 & theta.samps.1 < theta.samps.3), mean(theta.samps.2 < theta.samps.3 & theta.samps.3 < theta.samps.1), mean(theta.samps.3 < theta.samps.1 & theta.samps.1 < theta.samps.2), mean(theta.samps.3 < theta.samps.2 & theta.samps.2 < theta.samps.1)) row.names(comp.theta) <- c("$\\text{pr\\left(\\theta_1 < \\theta_2 < \\theta_3 \\right)$", "$\\text{pr\\left(\\theta_1 < \\theta_3 < \\theta_2 \\right)$", "$\\text{pr\\left(\\theta_2 < \\theta_1 < \\theta_3 \\right)$", "$\\text{pr\\left(\\theta_2 < \\theta_3 < \\theta_1 \\right)$", "$\\text{pr\\left(\\theta_3 < \\theta_1 < \\theta_2 \\right)$", "$\\text{pr\\left(\\theta_3 < \\theta_2 < \\theta_1 \\right)$") kable(comp.theta, digits = 2) Pr (θ 1 < θ 2 < θ 3 ) 0.01 Pr (θ 1 < θ 3 < θ 2 ) 0.00 Pr (θ 2 < θ 1 < θ 3 ) 0.08 Pr (θ 2 < θ 3 < θ 1 ) 0.67 Pr (θ 3 < θ 1 < θ 2 ) 0.02 Pr (θ 3 < θ 2 < θ 1 ) 0.22 Part c. comp.tilde.y <- rbind(mean(tilde.y.samps.1 < tilde.y.samps.2 & tilde.y.samps.2 < tilde.y.samps.3), mean(tilde.y.samps.1 < tilde.y.samps.3 & tilde.y.samps.3 < tilde.y.samps.2), mean(tilde.y.samps.2 < tilde.y.samps.1 & tilde.y.samps.1 < tilde.y.samps.3), mean(tilde.y.samps.2 < tilde.y.samps.3 & tilde.y.samps.3 < tilde.y.samps.1), mean(tilde.y.samps.3 < tilde.y.samps.1 & tilde.y.samps.1 < tilde.y.samps.2), mean(tilde.y.samps.3 < tilde.y.samps.2 & tilde.y.samps.2 < tilde.y.samps.1)) row.names(comp.tilde.y) <- c("$\\text{pr\\left(\\tilde{y_1 < \\tilde{y_2 < \\tilde{y_3 \\right)$", "$\\text{pr\\left(\\tilde{y_1 < \\tilde{y_3 < \\tilde{y_2 \\right)$", "$\\text{pr\\left(\\tilde{y_2 < \\tilde{y_1 < \\tilde{y_3 \\right)$", "$\\text{pr\\left(\\tilde{y_2 < \\tilde{y_3 < \\tilde{y_1 \\right)$", "$\\text{pr\\left(\\tilde{y_3 < \\tilde{y_1 < \\tilde{y_2 \\right)$", "$\\text{pr\\left(\\tilde{y_3 < \\tilde{y_2 < \\tilde{y_1 \\right)$") kable(comp.tilde.y, digits = 2) Pr (ỹ 1 < ỹ 2 < ỹ 3 ) 0.11 Pr (ỹ 1 < ỹ 3 < ỹ 2 ) 0.11 Pr (ỹ 2 < ỹ 1 < ỹ 3 )

4 Pr (ỹ 2 < ỹ 3 < ỹ 1 ) 0.27 Pr (ỹ 3 < ỹ 1 < ỹ 2 ) 0.14 Pr (ỹ 3 < ỹ 2 < ỹ 1 ) 0.20 Part d. comp <- rbind(mean(theta.samps.1 > theta.samps.2 & theta.samps.1 > theta.samps.3), mean(tilde.y.samps.1 > tilde.y.samps.2 & tilde.y.samps.1 > tilde.y.samps.3)) row.names(comp) <- c("$\\text{pr\\left(\\theta_1 > \\theta_2\\text{ \\& \\theta_1 > \\theta_3\\right)$ "$\\text{pr\\left(\\tilde{y_1 > \\tilde{y_2\\text{ \\& \\tilde{y_1 > \\tilde{y kable(comp, digits = 2) Pr (θ 1 > θ 2 & θ 1 > θ 3 ) 0.89 Pr (ỹ 1 > ỹ 2 & ỹ 1 > ỹ 3 ) 0.47 Problem 2 n.a <- n.b <- 16 y.bar.a < s.a <- 7.3 y.bar.b < s.b <- 8.1 mu.0 <- 75 sig.sq.0 <- 100 kappa.0 <- nu.0 <- 2^seq(0, 5, by = 1) kappa.n.a <- kappa.0 + n.a kappa.n.b <- kappa.0 + n.b mu.n.a <- (kappa.0*mu.0 + n.a*y.bar.a)/kappa.n.a mu.n.b <- (kappa.0*mu.0 + n.b*y.bar.b)/kappa.n.b nu.n.a <- nu.0 + n.a nu.n.b <- nu.0 + n.b sig.sq.n.a <- (nu.0*sig.sq.0 + (n.a - 1)*s.a^2 + ((kappa.0*n.a)/kappa.n.a)*(y.bar.a - mu.0)^2)/nu.n.a sig.sq.n.b <- (nu.0*sig.sq.0 + (n.b - 1)*s.b^2 + ((kappa.0*n.b)/kappa.n.b)*(y.bar.b - mu.0)^2)/nu.n.b ests <- numeric(length(kappa.0)) # Compute posterior theta_a < theta_b S < for (i in 1:length(ests)) { sigma.sq.inv.samps.a <- rgamma(s, nu.n.a[i]/2, nu.n.a[i]*sig.sq.n.a[i]/2) 4

5 theta.samps.a <- rnorm(s, mu.n.a[i], sqrt((1/sigma.sq.inv.samps.a)/kappa.n.a[i])) sigma.sq.inv.samps.b <- rgamma(s, nu.n.b[i]/2, nu.n.b[i]*sig.sq.n.b[i]/2) theta.samps.b <- rnorm(s, mu.n.b[i], sqrt((1/sigma.sq.inv.samps.b)/kappa.n.b[i])) ests[i] <- mean(theta.samps.a < theta.samps.b) plot(kappa.0, ests, xlab = expression(paste(kappa[0], ", ", nu[0], sep = "")), ylab = expression(paste("pr(", theta[a], "<", theta[b], " ", y[a], ", ", y[b], ")"), sep = "")) Pr(θ A <θ B y A, y B ) κ 0, ν 0 From the plot, we see that Pr (θ A < θ B y A, y B ) is not very sensitive to the choice of κ 0 and ν 0 when κ 0 = ν 0. As κ 0 and ν 0 increase, i.e. our prior belief that θ A = θ B and σ A = σ B increases in certainty, Pr (θ A < θ B y A, y B ) decreases slowly. Someone with a strong prior belief that θ A = θ B, i.e. κ 0 = ν 0 = 32, would still conclude that θ A < θ B after seeing the data. 5